Chapter 8 of 16
Syllabus
Time Series — Syllabus
Maharashtra Board · Class 12 · Mathematics & Statistics -Commerce
What Time Series covers in Maharashtra Board Class 12 Mathematics & Statistics -Commerce: 4 topics, for the 2026-27 session.
44 questions12 formulas & key relations5 concepts
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4 Topics · Maharashtra Board Class 12 Mathematics & Statistics -Commerce · 2026-27
Topics in Time Series
1
Definite Integral as Limit of Sum
- Definite integral ∫ᵇₐ f(x) dx represents the area under curve y = f(x) between x = a and x = b
- Derived from dividing interval [a,b] into n equal parts and taking limit as n → ∞
- Formula: ∫ᵇₐ f(x) dx = lim(n→∞) Σᵣ₌₁ⁿ h·f(a + rh) where h = (b-a)/n
2
Fundamental Theorem of Integral Calculus
- If F(x) is antiderivative of f(x), then ∫ᵇₐ f(x) dx = F(b) - F(a)
- Connects differentiation and integration as inverse operations
- Eliminates need for limit calculations in most practical problems
3
Properties of Definite Integrals - Basic Properties
- Property I: ∫ₐₐ f(x) dx = 0 (integral over zero width interval)
- Property II: ∫ᵇₐ f(x) dx = -∫ₐᵇ f(x) dx (swapping limits changes sign)
- Property III: ∫ᵇₐ f(x) dx = ∫ᵇₐ f(t) dt (integration variable doesn't matter)
4
Advanced Properties - Substitution Properties
- Property V: ∫ᵇₐ f(x) dx = ∫ᵇₐ f(a + b - x) dx (substitution property)
- Property VI: ∫₀ₐ f(x) dx = ∫₀ₐ f(a - x) dx (special case for [0,a])
- Property VII: ∫₀²ₐ f(x) dx = ∫₀ₐ f(x) dx + ∫₀ₐ f(2a - x) dx (interval doubling)
Key Concepts
The definite integral ∫ᵇₐ f(x)dxIf F(x) is an antiderivativeEight key properties simplify definite integralFor even functions f(For integrals of the form ∫₀^π/2
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What are the important topics in Time Series for Maharashtra Board Class 12 Mathematics & Statistics -Commerce?
Key topics in Time Series include Definite Integral as Limit of Sum, Fundamental Theorem of Integral Calculus, Properties of Definite Integrals - Basic Properties, Advanced Properties - Substitution Properties. Study these first, then practise questions on each for the Maharashtra Board Class 12 board exam.
How should I revise Time Series for the Maharashtra Board Class 12 board exam?
Learn the core ideas first, then work through the 44 practice questions on Time Series. Revise definitions regularly and use flashcards for quick recall before the exam.
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Content is aligned to the official syllabus. Refer to the board website for the latest curriculum.
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